In Section 3.3 of the main body of the paper, we introduced informative intra-cluster group (ICG) sizes. In such scenarios, there is an additional hierarchy given by the presence of known groups within the distinct clusters, and the size of the known groups may be related to the observed outcomes. Let \(i=1,...,m\)  index the clusters and let \(j=1,...,n_i\)  index the cluster units. Suppose each cluster comprises two known groups with at least one observation in each group. The methods described here can be easily modified to the cases where there are more than two known groups within a given cluster, and certain clusters may not have units belonging to any of the known groups. Suppose \(G_{ij}\)  is a subject-level binary level covariate indicating group membership, let \(n_{i0}\)  and \(n_{i1}\)  denote the respective group size corresponding to groups \(G_{i\cdot }=0\)  and \(G_{i\cdot }=1\)  in cluster \(i\) . Under this data structure, we explain the implementation of the pseudo-value regression for testing the hypothesis \(H_0: P\lbrace X_{ij}(t) = \ell | G_{ij} = 0, \mathbf {Z}_{ij}=\mathbf {z}\rbrace  = P\lbrace X_{ij}(t) = \ell | G_{ij} = 1, \mathbf {Z}_{ij}=\mathbf {z}\rbrace \)  for \(\ell \in \mathcal {S}\)  and \(t \ge 0\) . We have shown that correct inference based on the pseudo-value approach is driven by formulating the estimating equations with appropriate weights. For a marginal inference, we consider three possible weights to adjust for the informative ICG sizes: (i) \(w_{ij}=1\) , (ii) \(w_{ij}=\frac{1}{n_{i}}\) , and (iii) \(w_{ij}=\frac{1}{2n_{iG_{ij}}}\) , where \(n_i\)  is the number of units in cluster \(i\) , and \(n_{iG_{ij}}\)  denotes the number of units in cluster \(i\)  with the same group membership as unit \(j\) . The utilization of the weights in the estimating equations respectively correspond to the traditional GEE [1]}, the CWGEE [2]}, and the doubly-weighted estimating equation (DWGEE) [3]}. Using simulation studies, we investigate the appropriateness of these estimating equations based on pseudo-value responses for the analysis of clustered CS data with ICS.
